Modeling Crowd DynamicsBrent Morgan1, 2, Krista Parry1, 3, Andy Platta1, 3, Mitch Wilson1, 41Mathematics, 2Engineering Physics, 3Chemistry, 4Mechanical Engineering

- Project Description
- Goal: To accurately model the movement of individuals in a crowd out of a room through a single or multiple exits.
- Take into account different levels of urgency with regard to the context of the exit circumstances.
- Expand the model to include different parameters such as individuals pushing or shoving, as well as the possibility for injuries.
- Take into account human behavior in crowd situations such as maintaining a certain amount of personal space and increased urgency as proximity to the exit increases.
- Scientific Challenges & Potential Applications
- To accurately model an individual’s movement while taking into account a certain degree of intelligence and free will.
- Planning escape routes for buildings.
- Determining room capacities for building coding.

Results

Based on our implications of the Swarm and Kirchner models, we found that even with our normalized parameters Bij and Rij, we were able to recreate graphs and movement patterns as in the two comparable models. We were able to observe similar trends regarding parameter values, such as comparing ks and kd. We were also able to verify patterns relating to the injury threshold and the percentage of people that escape given an increasing number of initial individuals. See the accompanying figures for illustrations.

Figure 2: Series of screen shots at increasing time steps during one trial with Ks=3, Kd=2. Active agents denoted in blue, injured agents denoted in red.

- Published Models for Crowd Dynamics
- Simulation of Evacuation Processes Using a Bionics-Inspired Cellular Automaton Model for Pedestrian Dynamics, by Ansgar Kirchner:
- Introduces the Kirchner Field Model. This model simulates the evacuation of people from a room. People’s movements are determined probabilistically with the probability distribution depending on proximity to the door and the movement of other agents. We will refer to this model as the Kirchner model.
- Macroscopic Effects of Microscopic Forces Between Agents in Crowd Models, by Colin M. Henein:
- Implements many of the ideas used in the Kirchner Field Model but adds to it with the addition of a force field that simulates agents pushing. This allows the addition of injuries to the model which render agents immobile.We will refer to this model as the Swarm Force model.

Glossary of Technical Terms

Cellular Automata: A regular array of identical finite state automata whose next state is determined solely by their current state and the state of their neighbors.

Bosons:Elementary particles that give the agents information about their surroundings. These particles are dropped by the agents when they occupy a cell, and other individuals are attracted to these particles. This can be likened to ants leaving a pheromone trail to signal a pathway for others to follow.

Figure 4: Plot of number injured as a function of the injury threshold. Ks=3, Kd=3

Figure 3: Graphs of people remaining in room as a function of time. Kd, see key, Ks=3.

- Methodology
- Used method of direct computer simulation and cellular automata to construct a grid representation of a room with a single exit.
- Constructed a probability distribution of an individual’s 8 neighboring cells, based on factors such as the proximity to the door, and previous paths of individuals. (Equations shown on right).
- Determine which individual moves first at each time step based on a Force Determining equation which is inversely related to the distance to the door.
- Use concepts from Kirchner Field Model [4] and Swarm Force Model [1]; use a dynamic and static field to determine the individual’s most probable move and include injuries of individuals due to pushing or crowding.
- The static field will be inversely related to the distance to the door, thus the cell that is closer to the door will have a higher probability that the individual will move to it.
- The dynamic field keeps track of the bosons that individuals have left when previously occupying the cell. This raises the probability of an individual moving to the cell to simulate individuals following others if a path is successful.
- The dynamic and static fields are weighted differently based on proximity to the door.
- Pushing is allowed, and if an individual is pushed past the threshold, they will become permanently injured, indicated by a red dot in the following graphs.

Parameters

Pij: Probability of individual moving to cell (i, j)

N: Normalization coefficient

Bij: Normalized dynamic preference factor

ij: Occupancy of the cell (i,j)

kD: Dynamic field coefficient

kS: Static field coefficient

Rij: Normalized static preference factor

Dij:: Literature dynamic preference factor

Sij: Literature static preference factor

ij: Swarm Force parameter, based on occupancy

ij: Swarm Force parameter, based on walls

Fij: Force in determining movement order of individuals

(X, Y): Coordinates of the exit door

Equations

- Force Determining equation:

Figure 5: Plot of percentage of individuals leaving the room as a function of density of individuals in the room. Ks=3, Kd=2

- Modified Swarm Force equation:

- References
- Henin, C. M. & White, T. Macroscopic Effects of Microscopic Forces Between Agents in Crowd Models.Physica. A 373, 694-712 (2007).
- Weng, W.G., Pan, L.L., Shen, S.F., Yuan, H.Y. Small-grid Analysis of Discrete Model for Evacuation from a Hall. Physica. A 374, 821-826 (2007).
- Seyfried, A., Steffen, B., Lippert, T. Basics of Modelling the Pedestrian Flow.Physica A 368, 232-238 (2006).
- Kirchner, Ansgar & Schadschneider, Andreas. Simulation of Evacuation Processes Using a Bionics-Inspired Cellular Automaton Model for Pedestrian Dynamics. Physica. A 312, 260-276 (2002).

Figure 1: Method determining order of individual movement per time step, depending on the Force Determining equation.

- Literature Swarm Force equation:

- Acknowledgments
- This project was mentored by Jorge Ramirez, whose help is acknowledged with great appreciation.
- Support from a University of Arizona TRIF (Technology Research Initiative Fund) grant to J. Lega is also gratefully acknowledged.